I actually do think that it is important in some ways to LOTR's themes that Gollum does not get redeemed.
The first thing I want to say is that not every writing decision is strictly better or worse. If Gollum had redeemed himself, I think that would have been fine too. Different themes, but fine. But back to what actually happens.
I think, for many people who are both viewing fiction, and thinking about real life, that the world is divided into Good People and Bad People. Good People are, well, inherently good and well-meaning. They make mistakes, and even do bad things, but it's because their heart is in the right place, and they can always come back to the light. Bad People, meanwhile, are rotten to the core. Any action they take that is Good is basically self-serving to advance their position, and they can never make up for any sin they commit no matter how hard they try. (Thank you, Calvin.) For many people, a "good" redemption arc is when a Good Person has a redemption arc. After all, they were always well-meaning, and just needed to see the error of their ways. A "bad" redemption arc is when a Bad Person tries to redeem themselves. Obviously, people can be forgiven, but there are some things you just never come back from, that are signs of an inherently rotten and malign soul. There's no way anyone who does stuff like (FILL IN THE BLANK) can really redeem themselves.
I say all this, because I want to make clear that by all possible definitions, Gollum is one of The Bad Ones. Yes, his situation is pitiable in some ways, but as Tolkien himself pointed out: "The Ring was too strong for Smeagol but he would never have had to endure it if he had not already been a mean sort of thief." Smeagol's first act upon finding the Ring was to kill his own cousin; compare that with how long it took for Bilbo or Frodo or Sam to start feeling the effects of the Ring. Gollum literally eats babies, for crying out loud. From a very cynical point of view (one that I don't necessarily uphold), one could (one-sidedly) argue that his budding friendship with Frodo was merely finding a point of connection with another living soul for the first time in hundreds of years rather than any sort of true remorse, and look how easily that relationship soured (Gollum and Frodo stay cordial for less than half a book).
But (finally getting to my point), that is exactly why it's important that Frodo spares Gollum, gives him mercy, attempts to befriend this wretched creature. Another facet of the Good Person/Bad Person dichotomy is that it transforms morality from an "internal rule" to an "external action" - murder is an act that is really neither good nor inherently bad, instead murder is Bad when it happens to Good People and Good when it happens to Bad People. Thus, morality becomes a subjective and self-absorbed game more than a code to live life by - since of course the self becomes the arbiter of all definitions, and no act can be unjustified. Frodo spares Gollum NOT because deep down, Smeagol is really a Good Person led astray by understandable mistakes. He spares Gollum because taking pity and having mercy is the right thing to do no matter who the action is being done to. Notably, the first and only time Frodo curses Gollum - "if you touch me ever again you shall be cast into the fire" - is the moment where it is implied the Ring finally takes full hold on Frodo's psyche.
Frodo's mercy is mistaken and undeserved, and yet that's the point. Human charity isn't necessarily pragmatic. It's not just given to those who "deserve" it. As Gandalf says, "Many that live deserve death. And some that die deserve life. Can you give it to them? Then do not be too eager to deal out death in judgement. For even the very wise cannot see all ends."
